The unconstrained (first best) social planner solution is characterized by a tuple of variables {e, wH, wL, θ, U} satisfying condition wH = wL, the reservation utility
equation (4.1), the job creation conditionc = q(θ)JH(wH, wL), as well as equations
(a) and (b) below.
(a). The planner’s effort choice is given by:
dp′
(e)∆yυ′
(w) = C′
(e) (7.2)
(b). The risk adjusted Hosios surplus split:
JH(wH, wL) = 1 − ηq ηq R(wH, wL, U) υ′ w (7.3)
Proof: Appendix IV.
As follows from proposition 5 the unconstrained social planner optimally setswH =
wL. This means that the optimal bonus payment b is set to zero, implying income
insurance for workers against productivity shocks. Having guaranteed income stabil- ity for the employed population the social planner chooses an optimal effort level by maximizing the total surplus of a filled job. This is given by equation (7.2), where the left hand side of equation stands for the social gain of increasing the effort, while the right hand side can be interpreted as a marginal loss. The social lossC′(e) is directly
estimated in worker utility units, while the social gain is estimated as an increase in the expected productivity flowdp′(e)∆y multiplied by the respective shadow price of
an output unit, represented by the termυ′(w).
Notice that the optimal effort equation of a social planner (eq. 7.2) is different from the worker incentive compatibility constraint (4.2). Here the social cost of increas- ing effortC′(e) coincides with a private cost of the employee, however the social gain
dp′(e)∆yυ′(w) is generally different from a private gain of the employee, which can be
expressed asdp′(e)(υ(wH) − υ(wL)). Denote x
0 – solution to the following equality:
∆yυ′
(x0) = υ(wH) − υ(wL), x0 > wL
Then competitive search equilibrium with risk averse workers, unobserved effort and bonus payments entails a downward effort distortion with respect to the first best out-
come ifw < x0, otherwise ifw > x0 effort is biased upward. Intuitively, ifw is less
than x0, the shadow price of a single output unit is high, so that the social gain ex-
pressed in worker utility units is higher than the private gain of a worker, as a result the social planner will demand more effort from workers compared to the decentralized equilibrium with risk averse workers and unobserved effort values. The opposite holds whenw > x0, in this case the social gain converted into worker utility units is lower
than the private gain of a worker and therefore the first best effort level is lower than effort in a decentralized equilibrium.
In addition, it should also be noted that effort distortions are purely attributed to the risk aversion of workers. If workers are risk neutral, the social gain of exerting ef- fort expressed asdp′(e)∆y coincides with a private gain of a worker dp′(e)b. This is
the case because firms in a decentralized equilibrium optimally choose the maximum bonus payment valueb = ∆y (see proposition 1). This, however, does not imply that
the first best social planner solution with risk neutral workers may be decentralized by the market. The reason is that the maximum effort value is not compatible with a zero bonus payment when effort is unobserved.
Now consider the second best solution where the social planner is constrained by the information asymmetries arising from the unobserved worker effort choice. In this case the social planner is maximizing the present discounted value of a sum of utility flows of the unemployed and employed individuals with respect to the choice variables represented by a tuple {wH, wL, θ}. The objective function of a constrained social
planner becomes: max wL,wH,θ Z ∞ 0 e−rthuυ(z) + (1 − u)ˆυ(wL, wH)idt
where υ is given above. In addition, the worker incentive compatibility constraint˜
(equation 4.2), as well as the budget constraint of the social planner and the unemploy- ment dynamics equation should be fulfilled. The result of this optimization problem is summarized in proposition 6.
Proposition 6: Consider a social planner implementing a variable wage contract.
The constrained (second best) social planner solution is characterized by a tuple of variables {e, wH, wL, θ, U} satisfying the reservation utility equation (4.1), the job
(4.2), as well as the risk-sharing equation (4.8) and the rent-sharing equation (4.9). Therefore, competitive search equilibrium with bonus payments and unobserved effort is constrained efficient.
Proof: Appendix IV.
Proposition 6 characterizes the major properties of the constrained social planner solu- tion. It follows that the set of five equations describing the optimal solution of a social planner coincides with the set of equations in a decentralized competitive search equi- librium with unobserved effort and bonus payments. This means that the social planner will choose exactly the same optimal package of labour compensation(wL, wH) result-
ing in the same effort level of the employed and the same market tightness variableθ.
Therefore, it can be concluded that competitive search equilibrium with risk averse workers, bonus payments and unobserved effort is constrained efficient.
8
Conclusions
This paper develops a model of competitive search with risk averse workers in the presence of asymmetric information. Information asymmetries arise from the fact that workers possess private information about their effort choice on the job. The moral hazard problem within a match forces firms to use motivation devices such as the bonus pay in order to provide workers with the correct working incentives. This setup creates in a situation where the equilibrium labour contract entails both a hiring and a motivation wage premia. The hiring premium results from the rent-sharing incentive of firms ensuring them a sufficient job-filling rate, while the motivation premium results from the firm’s risk-sharing incentive necessary to guarantee a sufficient effort level.
The baseline model of the paper is compared to the classical model of moral haz- ard extended to account for labour market search frictions but preserving the essence of the ex-post wage setting mechanism. This benchmark model is proved to predict a lower amount of the bonus pay than the baseline model with wage competition be- tween employers. Similarly, both models are compared in the presence of a wage restriction imposed to reflect a binding limited liability constraint or a minimum wage requirement.
Furthermore, the paper presents an extension of the competitive search model with bonus payments to account for jobs heterogeneity. In particular, jobs are allowed to differ with respect to their capital endowments affecting both the expectation and the variation of output. The rent-sharing motive forces more capital intensive firms to leave higher rents to their employees. The higher rent comes in the form of a higher base wage as well as a higher bonus pay values. This complementarity effect provides ra- tionale for the positive cross-sectional correlation between bonus payments and wages reported in a number of empirical studies. The rent-sharing motive is absent in the model with an ex-post wage setting so that bonus payments and wages act as substi- tutes in a cross-section of firms. Based on the above theoretical analysis this paper concludes that the correlation between bonus payments and wages is specific to the type and the structure of the labour market. This is also in line with the observed em- pirical evidence.
Finally, this paper considers efficiency implications of incentive contracts in a com- petitive search equilibrium. The equilibrium is proved to be constrained efficient in the absence of tax payments and unemployment benefits. Nevertheless, competitive search equilibrium with bonus payments does not coincide with the full information allocation of the social planner. This is due to the fact that the private gain from ex- erting effort is different from the social gain, so that in the full information allocation the social planner will demand a different effort level from workers compared to the decentralized equilibrium.